DOCSLIB.ORG

Log-Concavity of Stirling Numbers and Unimodality of Stirling Distributions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Log-Concavity of Stirling Numbers and Unimodality of Stirling Distributions Ann. Inst. Statist. Math. Vol. 40, No. 4, 693 714 (1988) LOG-CONCAVITY OF STIRLING NUMBERS AND UNIMODALITY OF STIRLING DISTRIBUTIONS MASAAKI SIBUYA Department of Mathematics, Keio University, HiyoshL Yokohama 223, Japan (Received August 7, 1987; revised December 17, 1987) Abstract. A series of inequalities involving Stirling numbers of the first and second kinds with adjacent indices are obtained. Some of them show log-concavity of Stirling numbers in three different directions. The in- equalities are used to prove unimodality or strong unimodality of all the subfamilies of Stirling probability functions. Some additional applications are also presented. Key words and phrases: Inequalities, strong unimodality, total positivity 2, Stirling family of probability distributions. I. Introduction "Stirling family of distributions" is a collection of eight subfamilies of discrete probability distributions which involve Stirling numbers of the first and second kinds (Sibuya (1986)). The subfamilies are classified as shown in Table 2. Although the forms of their probability functions are different, all are unimodal if monotone increasing or decreasing probability functions in their distribution ranges are regarded also unimodal. This fact can be proved by log-concavity of Stirling numbers or by inequalities involving Stirling numbers with adjacent indices. Each of the first and second kinds of Stirling numbers forms a triangular array, which is totally positive 2 (Karlin (1968)) as a function of two indices. The array includes finite and infinite sequences in three directions. All the sequences are log-concave (Keilson and Gerber (1971)) to the extent that some are just as the definition, and some are stronger and one is weaker than the definition (cf. Table 1). Such type of inequalities was studied by Lieb (1968) and Neuman (1985). Results of this paper extend and improve some of their results. For some others alternative proofs are given. Lieb (1968) used Newton's inequality for symmetry functions and Neuman (1985) used moments of spline functions to obtain their inequalities. The methods used in this paper 693 694 MASAAKI SIBUYA are elementary. Log-concavity conditions are given by Kurtz (1972) for triangular arrays defined by a recurrence formula. In Section 2, preliminaries on Stirling numbers, Stirling distributions and log-concavity are given. In Section 3, a series of inequalities are given, and the main inequalities are summarized in Table 1. In Section 4, the inequalities are used to prove strong unimodality or unimodality of all the probability functions in Table 2 for any parameter value. The inequalities are applied to other problems (Section 5). One is Poisson upper bounds of the distributions of STR1F and STR2F subfamilies of Table 2. Another problem is monotonicity of the convolutions of Stirling numbers. Finally, log-concavity of binomial coefficients are remarked (Section 6). 2. Preliminaries 'Unsigned' Stirling numbers of the first kind [n] for nonnegative integers n and rn are defined by the polynomial identity t tn]:=t(t+l)...(t+n-l)=m E mrl ]tm ' in the domain 0 < m ___ n, and are zero outside the domain except that [0]=1 by convention. They satisfy the recurrence relation LVJ ~2,~ [n~,] =n [o][o]+ n>0, m>l m m m-1 ' - - " Stirling num~e.s of the secon~ ~in~ {~} for., m ~_ 0 a.e ~ef, ne~ similarly, in this case by where t/") := t(t - 1)...(t - m + 1). They satisfy the recurrence relation 22, =m {o}{n}+ n>0, m>l m m m- 1 ' - - " See Jordan (1947), Riordan (1958), Goldberg et al. (1964) and Knuth (1967-1981), for the introduction to Stirling numbers. The notation of the numbers differs by the literature; this paper follows the notation of Knuth. UNIMODALITY OF STIRLING DISTRIBUTIONS 695 Probability functions involving Stirling numbers as their main component are shown in Table 2, which appeared in Sibuya (1986). The table has two columns for each of two kinds of Stirling numbers. Good correspondence exists between them, and subfamilies have symbols STRI* in the first column, and STR2* in the second. The asterisks stand for F, W, C or I. The characters are initials of Finite range (or Fundamental), Waiting time, Complementary waiting time, and Infinite range. Some are very well known, some others are less known, and STR1W, STR1C and STR2C may be new. A series of nonnegative numbers (c,)n~-oo is log-concave if c,>_c,_~c,+~,2 n=O, +_ 1, _+2,..., (Hardy et al. (1952)). Log-concavity of a probability function on integers is a necessary and sufficient condition for its strong unimodality in Ibragimov's sense (Keilson and Gerber (1971)). Log-concavity of probability functions is preserved under the operations, convolution, shift, truncation and reverse. Strong unimodality implies unimodality, but not reversely. Log-concavity of the subfamilies STR1F, STR2W and STR2I is obvious. Their probability functions are multiple convolutions of Bernoulli, geometric and 0-truncated Poisson probability functions, respectively, and these are log-concave. The probability function of STRll is also a multiple convolution of logarithmic series probability functions which are, however, log-convex. 3. Inequalities In Theorems 3.1-3.7, inequalities on Stirling numbers are shown. The main inequalities in Theorems 3.1-3.5 show log-concavity in both kinds of Stirling numbers in three directions. The schemes of indices of Stirling numbers in the inequalities are illustrated in Table 1. A duality exists between the first and second kinds of Stirling numbers, and the same type of inequalities often holds for both kinds. Occasionally, however, sharp differences arise. The proofs for all the theorems, except for Theorem 3.7, are based on the recurrence relations (2.1) and (2.2) although some maneuvering is necessary. THEOREM 3.1. (Lieb (1968)) The following sequences are strictly decreasing for any n = 3, 4,... ; n-m+l m j m-1 ' m=2,3,...,n, 696 MASAAKI SIBUYA Table 1. Log-concavity and related inequality relations among Stifling numbers. the 1st Pattern the 2nd Symbol kind m-1 m m+l m+2 kind in the n- 1 Property text [n] n n+| A S s g2 s S log-concave g s totally B posmve s g s C W g2 log-concave s s G -- g2 -- log-concave s s g F S S -- g s s g D S S -- g s s g E S S g s 1. The central column, "pattern", indicates the positions of indices n and m appearing in each inequality. "g" represents the greater part and "s" the smaller part of the inequality. For example, in row A the pattern means En]2~[]m mn , ][ m+,n an~ /°/2~{~/m mn , J/ m+, 2. "S" in the columns on both sides indicates that the inequality is actually stronger than what the pattern appears, and some factor larger than 1 can be multiplied to the smaller side of the inequality. "W" in the first kind column means the contrary; the inequality is weaker, and some factor larger than 1 must be multiplied to the greater part. (AI') --mini { ° ] m= 1,2,...,n- 1 , ?'/--m /'/--m n-m+l ' {n} ,A2, m l,m{n} m = 2, 3,...,n, n-m+ l rn m-1 ' and > ..] > o b~ 0 I 0 [I f~ + + + c; I I iN) -] I f9 Eo I I I + o.q I I ~-~° + + o 0 t ~ r 0 I + + .-.. I I I '-' I + ,~ + II I © V I I I ,--t I + + f~ > + I i ! ! I i I ~ + i ! I t + ,.Qf~ + ,~ I ! = ,--t ~..~° 010 II + ! ! ! ,--t I + :Z i ! I II ,_, + + I i I 698 MASAAKI SIBUYA (3.2) ~m- ~,~o- m,{~}2 > (m + 1)(n- m + 1) {n }{n} m m-1 m+l ' m=2,...,n-1, which will be proved in a similar way. For n = 3 and m = 2, {3}2 {3}{3/6 2 =9>6 1 3 " To advance the induction step on n, calculate (m-1)(n-m+l){n+l}2-(m+l)(n-m+2){n+ll}{n+ll}m m- m+ =(m2-1) n-m+2((m-1)(n-m){n)2n-m+l m -(m+l)(n-m+l){m_ln }{m+ln }) + m -1 ( m2 + (n _ m + 2)(n _ m) ){ n }2 n-m+l m + -- m+'(~m2,~m~,~nm,{°m m- 1 }{nm - m(m + l)(n - m + 2) { n }{ n/) m-2 m +1 + --m -1 ( 2(n - 2) + (m - 2)2 ){ m n-1 }{ mn + m+l(-- (m-2)(n-m+ 1) {n}2 m m-1 m,. m+2,{ m~" }{:}) +2n-m+l{n} 2 m m-1 " All the terms are nonnegative and some are positive for m -- 2, 3,..., n. [] Remarks. Lieb (1968) showed (3.1) from Newton's inequality and the fact that the generating function of m m=~ has only real roots. He UNIMODALITY OF STIRLING DISTRIBUTIONS 699 showed also the sequence ,~, m, {nl/{. } n-m+l m-1 ' m=2,3,...,n, strictly decreasing. The inequality (A2) slightly improves this result. Numerically, it is suggested that the sequence (m,)(2n+m)[n]/[n ] (3.4) n-m+-I m, m-1 ' m=2,3,...,n-1, is strictly decreasing and the same for m = n - 1 and n, and that (3.5) n-m+l{m~)2 In}I{m j m-1n } ' m=2,3,...,n, is strictly decreasing. COROLLARY 3.1. Suppose a sequence (am)m:l satisfies 0 = a~ < a2 <-- a3 <- ... and 2am _> am-~ + am+l, m = 2, 3,.... The sequences am m = 2, 3,...,n, [nil[n]m m-1 ' am+ l in ]~[ I , m = 1, 2,...,n- 1 , {~}/{nn-m ~ n-m+l} am m = 2, 3,...,n, m m-1 ' and {n}/{ n } am+l m = 1, 2,...,n - 1 n-m n-m+l ' are strictly decreasing for any n = 3, 4,...
Load more

Recommended publications
  • Exploring Bimodality in Introductory Computer Science Performance Distributions
    EURASIA Journal of Mathematics, Science and Technology Education, 2018, 14(10), em1591 ISSN:1305-8223 (online) OPEN ACCESS Research Paper https://doi.org/10.29333/ejmste/93190 Exploring Bimodality in Introductory Computer Science Performance Distributions Ram B Basnet 1, Lori K Payne 1, Tenzin Doleck 2*, David John Lemay 2, Paul Bazelais 2 1 Colorado Mesa University, Grand Junction, Colorado, USA 2 McGill University, Montreal, Quebec, CANADA Received 7 April 2018 ▪ Revised 19 June 2018 ▪ Accepted 21 June 2018 ABSTRACT This study examines student performance distributions evidence bimodality, or whether there are two distinct populations in three introductory computer science courses grades at a four-year southwestern university in the United States for the period 2014-2017. Results suggest that computer science course grades are not bimodal. These findings counter the double hump assertion and suggest that proper course sequencing can address the needs of students with varying levels of prior knowledge and obviate the double-hump phenomenon. Studying performance helps to improve delivery of introductory computer science courses by ensuring that courses are aligned with student needs and address preconceptions and prior knowledge and experience. Keywords: computer science performance, coding, programming, double hump, grade distribution, bimodal distribution, unimodal distribution BACKGROUND In recent years, there has been a resurgence of interest in the practice of coding (Kafai & Burke, 2013), with many pushing for making it a core competency for students (Lye & Koh, 2014). There are inherent challenges in learning to code evidenced by high failure and dropout rates in programming courses (Ma, Ferguson, Roper, & Wood, 2011; (Qian & Lehman, 2017; Robins, 2010).
    [Show full text]
  • Recombining Partitions Via Unimodality Tests
    Working Paper 13-07 Departamento de Estadística Statistics and Econometrics Series (06) Universidad Carlos III de Madrid March 2013 Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624-98-48 RECOMBINING PARTITIONS VIA UNIMODALITY TESTS Adolfo Álvarez (1) , Daniel Peña(2) Abstract In this article we propose a recombination procedure for previously split data. It is based on the study of modes in the density of the data, since departing from unimodality can be a sign of the presence of clusters. We develop an algorithm that integrates a splitting process inherited from the SAR algorithm (Peña et al., 2004) with unimodality tests such as the dip test proposed by Hartigan and Hartigan (1985), and finally, we use a network configuration to visualize the results. We show that this can be a useful tool to detect heterogeneity in the data, but limited to univariate data because of the nature of the dip test. In a second stage we discuss the use of multivariate mode detection tests to avoid dimensionality reduction techniques such as projecting multivariate data into one dimension. The results of the application of multivariate unimodality tests show that is possible to detect the cluster structure of the data, although more research can be oriented to estimate the proper fine-tuning of some parameters of the test for a given dataset or distribution. Keywords: Cluster analysis, unimodality, dip test. (1) Álvarez, Adolfo, Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail: [email protected] . (2) Peña, Daniel, Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail: [email protected] .
    [Show full text]
  • UNIMODALITY and the DIP STATISTIC; HANDOUT I 1. Unimodality Definition. a Probability Density F Defined on the Real Line R Is C
    UNIMODALITY AND THE DIP STATISTIC; HANDOUT I 1. Unimodality Definition. A probability density f defined on the real line R is called unimodal if the following hold: (i) f is nondecreasing on the half-line ( ,b) for some real b; (ii) f is nonincreasing on the half-line (−∞a, + ) for some real a; (iii) For the largest possible b in (i), which∞ exists, and the smallest possible a in (ii), which also exists, we have a b. ≤ Here a largest possible b exists since if f is nondecreasing on ( ,bk) −∞ for all k and bk b then f is also nondecreasing on ( ,b). Also b < + since f is↑ a probability density. Likewise a smallest−∞ possible a exists∞ and is finite. In the definition of unimodal density, if a = b it is called the mode of f. It is possible that f is undefined or + there, or that f(x) + as x a and/or as x a. ∞ ↑ ∞ If a<b↑ then f is constant↓ on the interval (a,b) and we can take it to be constant on [a,b], which will be called the interval of modes of f. Any x [a,b] will be called a mode of f. When a = b the interval of modes∈ reduces to the singleton a . { } Examples. Each normal distribution N(µ,σ2) has a unimodal density with a unique mode at µ. Each uniform U[a,b] distribution has a unimodal density with an interval of modes equal to [a,b]. Consider a−1 −x the gamma density γa(x) = x e /Γ(a) for x > 0 and 0 for x 0, where the shape parameter a > 0.
    [Show full text]
  • Field Guide to Continuous Probability Distributions
    Field Guide to Continuous Probability Distributions Gavin E. Crooks v 1.0.0 2019 G. E. Crooks – Field Guide to Probability Distributions v 1.0.0 Copyright © 2010-2019 Gavin E. Crooks ISBN: 978-1-7339381-0-5 http://threeplusone.com/fieldguide Berkeley Institute for Theoretical Sciences (BITS) typeset on 2019-04-10 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 271828182845904 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probabil- ity distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of de- scribing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diver- sity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be orga- nized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hun- dred or so simple distributions, their properties and their interrelations.
    [Show full text]
  • On Multivariate Unimodal Distributions
    ON MULTIVARIATE UNIMODAL DISTRIBUTIONS By Tao Dai B. Sc. (Mathematics), Wuhan University, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF STATISTICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1989 © Tao Dai, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of £ tasW> tj-v*> The University of British Columbia Vancouver, Canada Date ./ v^y^J. DE-6 (2/88) ABSTRACT In this thesis, Kanter's representation of multivariate unimodal distributions is shown equivalent to the usual mixture of uniform distributions on symmetric, compact and convex sets. Kanter's idea is utilized in several contexts by viewing multivariate distributions as mixtures of uniform distributions on sets of various shapes. This provides a unifying viewpoint of what is in the literature and .gives some important new classes of multivariate unimodal distributions. The closure properties of these new classes under convolution, marginality and weak convergence, etc. and their relationships with other notions of multivariate unimodality are discussed. Some interesting examples and their 2- or 3-dimensional pictures are presented.
    [Show full text]
  • Large Time Unimodality for Classical and Free Brownian Motions with Initial Distributions
    ALEA, Lat. Am. J. Probab. Math. Stat. 15, 353–374 (2018) DOI: 10.30757/ALEA.v15-15 Large time unimodality for classical and free Brownian motions with initial distributions Takahiro Hasebe and Yuki Ueda Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan E-mail address: [email protected] URL: https://www.math.sci.hokudai.ac.jp/~thasebe/ Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan E-mail address: [email protected] Abstract. We prove that classical and free Brownian motions with initial distri- butions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar re- sults are shown for a symmetric stable process with index 1 and a positive stable process with index 1/2. We also prove that free Brownian motion with initial sym- metric unimodal distribution is unimodal, and discuss strong unimodality for free convolution. 1. Introduction A Borel measure µ on R is unimodal if there exist a R and a function f : R [0, ) which is non-decreasing on ( ,a) and non-increasing∈ on (a, ), such that→ ∞ −∞ ∞ µ(dx)= µ( a )δ + f(x) dx. (1.1) { } a The most outstanding result on unimodality is Yamazato’s theorem (Yamazato, 1978) saying that all classical selfdecomposable distributions are unimodal. After this result, in Hasebe and Thorbjørnsen (2016), Hasebe and Thorbjørnsen proved the free analog of Yamazato’s result: all freely selfdecomposable distributions are unimodal.
    [Show full text]
  • Nber Working Paper Series the Distribution of the Size
    NBER WORKING PAPER SERIES THE DISTRIBUTION OF THE SIZE OF PRICE CHANGES Alberto Cavallo Roberto Rigobon Working Paper 16760 http://www.nber.org/papers/w16760 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 February 2011 We are grateful to Alberto Alesina, Emmanuel Farhi, Oleksiy Kryvtsov, Greg Mankiw, Dan Sichel, and seminar participants at Harvard, MIT, NBER, AEA, BLS, ITAM, Oxford, Queen’s University at Belfast, and the Central Bank of Norway for helpful comments and suggestions. All remaining errors are ours. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer- reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2011 by Alberto Cavallo and Roberto Rigobon. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. The Distribution of the Size of Price Changes Alberto Cavallo and Roberto Rigobon NBER Working Paper No. 16760 February 2011 JEL No. E00,E3 ABSTRACT Different theories of price stickiness have distinct implications on the properties of the distribution of price changes. One of those characteristics is the number of modes in the distribution. We formally test for the number of modes in the price change distribution of 32 supermarkets, spanning 23 countries and 5 continents. We present results for three modality tests: the two best-known tests in the statistical literature, Hartigan’s Dip and Silverman’s Bandwith, and a test designed in this paper, called the Proportional Mass test (PM).
    [Show full text]
  • The Basic Distributional Theory for the Product of Zero Mean Correlated
    The basic distributional theory for the product of zero mean correlated normal random variables Robert E. Gaunt∗ Abstract The product of two zero mean correlated normal random variables has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to tabulated. This paper fills this gap by providing the basic distributional theory for the product of two zero mean correlated normal random variables. Keywords: Product of correlated normal random variables; distributional theory AMS 2010 Subject Classification: Primary 60E05; 62E15 1 Introduction 2 2 Let (X,Y ) be a bivariate normal random vector with zero mean vector, variances (σX , σY ) and correlation coefficient ρ. The distribution of the product Z = XY arises in nu- merous applications, with recent examples including statistics of Lagrangian power in two-dimensional turbulence (Bandi and Connaughton [3]), product confidence limits for indirect effects (MacKinnon et al. [28]) and statistical mediation analysis (MacKinnon −1 [27]). The mean Zn = n (Z1 + Z2 + + Zn), where Z1,Z2,...,Zn are independent and identical copies of Z, has also found··· application in areas such as electrical engineering (Ware and Lad [42]), astrophysics (Mangilli, Plaszczynski and Tristram [29], Watts et al. [43]) and quantum cosmology (Grishchuk [18]). The exact distribution of the product Z = XY has been studied since 1936 (Craig arXiv:2106.02897v1 [math.ST] 5 Jun 2021 [5]), with subsequent contributions coming from Aroian [1], Aroian, Taneja and Cornwell [2], Bandi and Connaughton [3], Haldane [20], Meeker et al. [31]; see Nadarajah and Pog´any [32] for an overview of these and further contributions.
    [Show full text]
  • Distributionally Robust Chance Constrained Optimal Power Flow Assuming Log-Concave Distributions
    Distributionally Robust Chance Constrained Optimal Power Flow Assuming Log-Concave Distributions B. Li 1, R. Jiang 2 and J.L. Mathieu 1 1EECS, University of Michigan 2IOE, University of Michigan November 2016 Bowen Li University of Michigan November 2016 1 / 17 Background: Chance Constrained Optimization Widely used in different problems with uncertainty But hard to accurately estimate uncertainty distributions with historical data Example: Physical constraints should be satisfied for most of the possible realizations of renewable/load uncertainty in power systems planning problems Solving methodologies Scenario approach: large number of scenarios/constraints Probabilistically robust method: overconservativeness Analytical reformulation: case dependent reliability General formulation P(L(x; ξ) ≤ 0) ≥ 1 − Bowen Li University of Michigan November 2016 2 / 17 Background: Distributionally Robust Optimization [1] Key properties A probabilistic constraint is satisfied within an ambiguity set of distributions The ambiguity set can be calibrated using the following information Moment-based information: mean, covariance, higher-order moments Density-based information: likelihood of a known probability density Distribution structure: unimodality, support, smoothness, symmetry Good trade-off between performance and computational tractability Example: distributionally robust chance constraint (DRCC) inf (L(x; ξ) ≤ 0) ≥ 1 − U P P2P U n T o P = Unimodal; EP(ξ) = µ, EP(ξξ ) = S [1] E. Delage and Y. Ye, \Distributionally Robust Optimization under Moment
    [Show full text]
  • Alpha-Skew Generalized T Distribution
    Revista Colombiana de Estadística July 2015, Volume 38, Issue 2, pp. 353 to 370 DOI: http://dx.doi.org/10.15446/rce.v38n2.51665 Alpha-Skew Generalized t Distribution Distribución t generalizada alfa sesgada Sukru Acitas1;a, Birdal Senoglu2;b, Olcay Arslan2;c 1Department of Statistics, Faculty of Science, Anadolu University, Eskisehir, Turkey 2Department of Statistics, Faculty of Science, Ankara University, Ankara, Turkey Abstract The alpha-skew normal (ASN) distribution has been proposed recently in the literature by using standard normal distribution and a skewing ap- proach. Although ASN distribution is able to model both skew and bimodal data, it is shortcoming when data has thinner or thicker tails than normal. Therefore, we propose an alpha-skew generalized t (ASGT) by using the gen- eralized t (GT) distribution and a new skewing procedure. From this point of view, ASGT can be seen as an alternative skew version of GT distribution. However, ASGT differs from the previous skew versions of GT distribution since it is able to model bimodal data sest as well as it nests most commonly used density functions. In this paper, moments and maximum likelihood estimation of the parameters of ASGT distribution are given. Skewness and kurtosis measures are derived based on the first four noncentral moments. The cumulative distribution function (cdf) of ASGT distribution is also ob- tained. In the application part of the study, two real life problems taken from the literature are modeled by using ASGT distribution. Key words: Bimodality, Kurtosis, Maximum Likelihood Estimation, Mod- eling, Skewness. Resumen La distribución normal alfa-sesgada (ASN por sus siglas en inglés) ha sido propuesta recientemente en la literatura mediante el uso de una distribución normal estándar y procedimientos de sesgo.
    [Show full text]
  • The Modes of a Mixture of Two Normal Distributions
    Silesian J. Pure Appl. Math. vol. 6, is. 1 (2016), 59–67 Grzegorz SITEK Department of Statistics, University of Economics in Katowice, Poland THE MODES OF A MIXTURE OF TWO NORMAL DISTRIBUTIONS Abstract. Mixture distributions arise naturally where a statistical pop- ulation contains two or more subpopulations. Finite mixture distributions refer to composite distributions constructed by mixing a number K of com- ponent distributions. The first account of mixture data being analyzed was documented by Pearson in 1894. We consider the distribution of a mix- ture of two normal distributions and investigate the conditions for which the distribution is bimodal. This paper presents a procedure for answer- ing the question of whether a mixture of two normal distributions which five known parameters µ1, µ2, σ1, σ2, p is unimodal or not. For finding the modes, a simple iterative procedure is given. This article presents the possibility of estimation of modes using biaverage. 1. Introduction Mixture models have been widely used in econometrics and social science, and the theories for mixture models have been well studied Lindsay (see [5]). The importance of the research for unimodality or bimodality in statistics have been described by Murphy (see [6]). Consider f(x, p)= pf (x)+(1 p)f (x) (1) 1 − 2 2010 Mathematics Subject Classification: 60E05, 62E99. Keywords: mixture of normal distributions, bimodal distributions, biaverage. Corresponding author: G. Sitek ([email protected]). Received: 25.07.2016. 60 G. Sitek where, for i =1, 2, 1 − − 2 2 (x µi) /2σi fi(x)= e σi√2π with 0 <p< 1. The function f(x, p) is the probability density function of a mixture of two normal distributions.
    [Show full text]
  • Distributionally Robust Chance Constrained Optimal Power
    1 Distributionally Robust Chance Constrained Optimal Power Flow Assuming Unimodal Distributions with Misspecified Modes Bowen Li, Student Member, IEEE, Ruiwei Jiang, Member, IEEE, and Johanna L. Mathieu, Member, IEEE Abstract—Chance constrained optimal power flow (CC-OPF) approximation [8], [9], which provides a priori guarantees formulations have been proposed to minimize operational costs on reliability, require the constraints to be satisfied over a while controlling the risk arising from uncertainties like renew- large number of uncertainty samples. The solutions from these able generation and load consumption. To solve CC-OPF, we often need access to the (true) joint probability distribution of approaches are usually overly conservative with high costs all uncertainties, which is rarely known in practice. A solution [7], [10]. Another popular approach is to assume that the based on a biased estimate of the distribution can result in uncertainties follow a parametric distribution such as Gaussian poor reliability. To overcome this challenge, recent work has [4], [5], [7]. The resulting CC-OPF is often easier to solve explored distributionally robust chance constraints, in which the but the solution may have low reliability unless the assumed chance constraints are satisfied over a family of distributions called the ambiguity set. Commonly, ambiguity sets are only probability distribution happens to be close to the true one. based on moment information (e.g., mean and covariance) of the As an alternative, distributionally robust chance constrained random variables; however, specifying additional characteristics (DRCC) OPF models do not depend on a single estimate of the random variables reduces conservatism and cost. Here, we of the probability distribution [10]–[17].
    [Show full text]